This paper is concerned with a special but important case of dimensionality reduction.
We assume that there is some unknown connected, compact smooth \(q\) -submanifold \(M\subset \mathbb {R}^p\)  for which \(X_1,\ldots ,X_n\)  are drawn from \(M\)  (\(\phi (x)=x\)  for all \(x\in M\) ).
This is the case, for example, in certain continuous latent space models of networks (eg. [1]}.)
This is also the case, for example, in the data of radar bearings (eg. [2]}.)
